Finally, we consider the eigenvalues of Wilson-loop operator as another property of the Hilbert space of the narrow bands. This operator \(\hat{O}\)  is defined as \(P e^{-i \frac{1}{N}_1 g_1 \cdot r} P\)  where \(N_1\)  is the number of unit cells along the direction of \(L_1\)  in the entire lattice with periodic boundary conditions, and \(P\)  is the projection operator onto the Bloch states of the narrow bands. Since this operator commutes with the momentum operator along \(g_2\) , its eigenstate is labeled by the momentum \(k\)  along \(g_2\) . In the BM model, the phase of the eigenvalues of \(\hat{O}\) , labeled as \(\langle x_{\pm } \rangle \) , has the winding number of \(\pm 1\)  as \(k\)  runs from 0 to 1, illustrating the nontrivial topological properties of the narrow band system [1]}, [2]}, [3]}.
